Are all topological graphs geometric graphs?

Question

A topological graph or string graph is an intersection graph of curves. Can all such curves be drawn as intersection graph of line segments?

Answer 1

Assuming my comment is correct, the answer is no. Consider a graph with vertices $a,b,c,d,e,f,g, h$ with edges connecting $a, b$ to all every vertex, and the following additional edges: $$ce, df, cg, dh$$

I claim this graph cannot be the intersection graph of a collection of lines. Assume otherwise. Lines $a$ and $b$ intersect in some point $P$ and determine a plane $\pi$. lines $c,d,e,f$ each either intersects $a$ and $b$ in the same point $P$, or else intersects them in different points. In the latter case, the line must lie in $\pi$.

If neither $c$ nor $d$ pass through $P$, then they lie in $\pi$. Since they do not intersect, they must be parallel. But then $e$ contains points in $a, b, c$. These cannot all be the same point, since $a$ and $b$ only intersect in $P$, and $c$ does not contain $P$ by the assumption. So $e$ passes through two distinct points of $\pi$ and therefore lies within it. $e$ does not not intersect $d$ and so must be parallel to it, and therefore to $c$. But $c$ and $e$ intersect, a contradiction.

So either $c$ or $d$ passes through $P$. By swapping $(c,e,g)$ with $(d,f,h)$ if needed, we can assume WLOG that $P \in c$. Since neither intersects $c$, it must be that $d,f$ both lie in $\pi$. If either of $c, e$ lies in $\pi$, it is parallel to both $d$ and $f$, so those two must be parallel as well. But they intersect, a contradiction. So both $c$ and $e$ do not lie in $\pi$ and only intersect it in $P$.

$g$ intersects with all of $a, b, c$. If it did not pass through $P$, then it would contain separate points from $a,b$ and thus would lie in $\pi$. But that prevents it from intersecting $c$, a contradiction. So $g$ passes through $P$. But this is impossible, since $P\in e$, but $g$ does not intersect $e$.

However, this same graph is easily accomplished as the intersection set of a collection of curves in space, where $a$ though $f$ can be lines, but $g$ a curve that stays out of the plane except to pass through a point each of $a$ and $b$, and which intersects $c$ at some point other than $P$.